HOMOGENEOUS COORDINATES AND QUOTIENT
PRESENTATIONS FOR TORIC VARIETIES
ANNETTE A’CAMPO-NEUEN, JÜRGEN HAUSEN, AND STEFAN SCHRÖER
Abstract. Generalizing cones over projective toric varieties, we present arbitrary toric varieties as quotients of quasiaffine toric varieties. Such quotient
presentations correspond to groups of Weil divisors generating the topology.
Groups comprising Cartier divisors define free quotients, whereas Q-Cartier
divisors define geometric quotients. Each quotient presentation yields homogeneous coordinates. Using homogeneous coordinates, we express quasicoherent
sheaves in terms of multigraded modules and describe the set of morphisms
into a toric variety.
1. Introduction
n
The projective space P is the quotient of the pointed affine space An+1 \ 0 by
the diagonal Gm -action. A natural question to ask is whether this generalizes to
other toric varieties. Indeed: Cox [4] and others showed that each toric variety X
is the quotient of a smooth quasiaffine toric variety X̂.
This quasiaffine toric variety X̂ and the corresponding homogeneous coordinate
ring Γ(X̂, OX̂ ), however, are very large and entail redundant information. For
toric varieties with enough invariant Cartier divisors, Kajiwara [7] found smaller
homogeneous coordinate rings.
The goal of this paper is to generalize homogeneous coordinates and to study
them from a geometric viewpoint. In our language, homogeneous coordinates correspond to quotient presentations. Both the constructions of Cox and Kajiwara
are quotient presentations; other examples are cones over quasiprojective toric varieties. Given any particular toric variety, our approach provides flexibility in the
choice of homogeneous coordinate rings.
Roughly speaking, a quotient presentation for a toric variety X is a quasiaffine
toric variety X̂, together with an affine surjective toric morphism q : X̂ → X such
that groups of invariant Weil divisors on X and X̂ coincide. The global sections
S = Γ(X̂, OX̂ ) are the corresponding homogeneous coordinates for X.
Homogeneous coordinates are useful for various purposes. For example, Cox
[3] described the set of morphism r : Y → X from a scheme Y into a smooth
toric variety X in terms of homogeneous coordinates. Subsequently, Kajiwara [7]
generalized this to toric varieties with enough effective Cartier divisors. Using homogeneous coordinates, Brion and Vergne [2] determined Todd classes on simplicial
toric varieties. Eisenbud, Mustata and Stillman [5] recently applied homogeneous
coordinates to calculate cohomology groups of coherent sheaves.
This article is divided into five sections. In the first section, we define the concept
of quotient presentations and give a characterization in terms of fans. Section 2
1991 Mathematics Subject Classification. 14M25, 14C20, 14L30, 14L32.
1
2
A. A’CAMPO-NEUEN, J. HAUSEN, AND S. SCHROEER
contains a description of quotient presentations in terms of groups of Weil divisors.
Such groups of Weil divisors are not arbitrary. Rather, they generalize the concept
of an ample invertible sheaf or an ample family of sheaves.
In Section 3, we relate quotient presentations to geometric invariant theory. Quotient presentations defined by Cartier or Q-Cartier divisors are free or geometric
quotients, respectively. Because quotients for group actions tend to be nonseparated, it is natural (and requires no extra effort) to consider nonseparated toric
prevarieties as well.
In Section 4, we shall express quasicoherent sheaves on toric varieties in terms
of multigraded modules over homogeneous coordinate rings. In the last section, we
describe the functor hX (Y ) = Hom(Y, X) represented by a toric variety X in terms
of sheaf data on Y related to homogeneous coordinates.
2. Quotient presentations
Throughout we shall work over an arbitrary ground field k. A toric variety is
an equivariant torus embedding T ⊂ X, where X is a separated normal algebraic
k-variety. As usual, N denotes the lattice of 1-parameter subgroups of the torus T ,
and M is the dual lattice of characters. Recall that toric varieties correspond to
finite fans ∆ in the lattice N . We shall encounter toric prevarieties as well: These
are equivariant torus embeddings as above, but with X possibly nonseparated.
Let q : X̂ → X be a surjective toric morphism of toric prevarieties. Then we
have a pullback homomorphism q ∗ : CDivT (X) → CDivT̂ (X̂) for invariant Cartier
divisors. There is also a strict transform for invariant Weil divisors defined as
follows. Let U ⊂ X be the union of all T -orbits of codimension ≤ 1, and Û ⊂ X̂
its preimage. Each invariant Weil divisor on X becomes Cartier on U , and the
composition
q∗
WDivT (X) = CDivT (U ) −→ CDivT̂ (Û ) ⊂ WDivT̂ (Û ) ⊂ WDivT̂ (X̂)
defines the strict transform q ] : WDivT (X) → WDivT̂ (X̂) on the groups of invariant
Weil divisors. Note that q ] is injective.
Definition 2.1. A quotient presentation for a toric prevariety X is a quasiaffine
toric variety X̂, together with a surjective affine toric morphism q : X̂ → X such
that the strict transform q ] : WDivT (X) → WDivT̂ (X̂) is bijective.
This notion is local: Given that X̂ is quasiaffine, a toric morphism q : X̂ → X is
a quotient presentation if and only if for each invariant affine open subset U ⊂ X
the induced toric morphism q −1 (U ) → U is a quotient presentation.
Example 2.2. The cones R+ (1, 0), and R+ (0, 1) in the lattice N̂ = Z2 define the
quasiaffine toric variety X̂ = A2 \ 0. The projection Z2 → Z2 /Z(1, 1) yields a
quotient presentation q : A2 \ 0 → X for the projective line X = P1 . We could
use the projection onto Z2 /Z(1, −1) as well. This defines a quotient presentation
q : A2 \ 0 → X for the affine line X = A1 ∪ A1 with origin doubled, which is a
nonseparated toric prevariety.
Here comes a characterization of quotient presentations in terms of fans. For
simplicity, we are content with the separated case. Suppose that q : X̂ → X is a
ˆ → (N, ∆).
toric morphism of toric varieties given by a map of fans Q : (N̂ , ∆)
QUOTIENT PRESENTATIONS
3
Theorem 2.3. The toric morphism q : X̂ → X is a quotient presentation if and
only if the following conditions hold:
(i) The lattice homomorphism Q : N̂ → N has finite cokernel.
ˆ is a subfan of the fan of faces of a strongly convex cone σ̄ ⊂ N̂R .
(ii) The fan ∆
ˆ max → ∆max and ∆
ˆ (1) →
(iii) The assignment σ 7→ QR (σ) defines bijections ∆
(1)
∆ .
ˆ the image
(iv) For each primitive lattice vector v̂ ∈ N̂ generating a ray ρ̂ ∈ ∆,
Q(v̂) ∈ N is a primitive lattice vector.
Proof. Suppose the conditions hold. The cone σ̄ ⊂ N̂R yields a toric open embedding X̂ ⊂ Xσ̄ , hence X̂ is quasiaffine.
To see that the map q : X̂ → X is surjective, consider an affine chart Xσ ⊂ X,
where σ ∈ ∆ is a maximal cone. Since Q induces a bijection of maximal cones,
ˆ max such that QR (σ̂) = σ. Moreover, Q was assumed to have a finite
there is a σ̂ ∈ ∆
cokernel, so q : T̂ → T is surjective. Since q is equivariant,this implies Xσ = q(Xσ̂ ).
To check that the map q : X̂ → X is affine, keep on considering Xσ . It is easy
to see that the inverse image of Xσ is
[
Xτ̂ .
(2.3.1)
q −1 (Xσ ) =
ˆ QR (τ̂ )⊂σ
τ̂ ∈∆;
ˆ (1) → ∆(1) we see that Q−1 (σ) contains no element of ∆
ˆ (1) \
Using the bijection ∆
R
ˆ mapped by QR into σ are the faces of σ̂.
σ̂ (1) . Consequently, the only cones of ∆
−1
By the above formula, this means q (Xσ ) = Xσ̂ . So we see that q : X̂ → X is
affine.
It remains to show that the strict transform is bijective. As to this, recall first
that the invariant prime divisors of X are precisely the closures of the T -orbits
Speck[ρ⊥ ∩ M ] ⊂ X where ρ ∈ ∆(1) .
We calculate the strict transform of a T -stable prime divisor D ⊂ X correspondˆ (1) → ∆(1) is bijective, there is a unique ray ρ̂ ∈ ∆
ˆ (1)
ing to a ray ρ ∈ ∆(1) . Since ∆
]
with QR (ρ̂) = ρ. It follows from 2.3.1 that q (D) is a multiple of the T̂ -invariant
prime divisor D̂ corresponding to ρ̂. Note that q −1 (Xρ ) = Xρ̂ . To calculate the
multiplicity of D̂ in q ] (D), it suffices to determine the pullback of D ∈ CDivT (Xρ )
via q : Xρ̂ → Xρ .
On the affine chart Xρ , every invariant Cartier divisor is principal, and if v is the
primitive lattice vector in ρ then the assignment m 7→ hm, viD induces a natural
isomorphism M/ρ⊥ ' CDivT (Xρ ). Since we have
q ∗ (div(χm )) = div(χm ◦ q) = div(χm◦Q ),
the pullback q ∗ : CDivT (Xρ ) → CDivT̂ (Xρ̂ ) corresponds to the map Q∗ : M/ρ⊥ →
M̂ /ρ̂⊥ . By condition (iv), this map is an isomorphism and hence q ∗ (D) = D̂.
ˆ (1) → ∆1 , you conclude that the strict transform is
Again using bijectivity of ∆
bijective. Thus the conditions are sufficient. Using similar arguments, you see that
the conditions are also necessary.
¤
Example 2.4. Suppose σ ⊂ R3 be a strongly convex cone generated by four
ˆ be the fan of all
extremal rays R+ v1 , . . . , R+ v4 , defining a fan ∆ in N = Z3 . Let ∆
faces of the first quadrant in N̂ = Zv1 ⊕ . . . ⊕ Zv4 . Then the canonical surjection
Q : N̂ → N gives a quotient presentation.
4
A. A’CAMPO-NEUEN, J. HAUSEN, AND S. SCHROEER
Q
Figure 1. A quotient presentation of a non simplicial affine toric variety
ˆ comprises
The induced map on nonzero cones looks like Figure 1. The fan ∆
16 cones, whereas ∆ contains only 10 cones. You see that two 2-dimensional cones
ˆ map to the maximal cone in ∆.
and all 3-dimensional cones in ∆
Example 2.5. Let ∆ be a polytopal fan in the lattice N , and for each ray ρ ∈ ∆(1)
let vρ ∈ ρ be the primitive lattice vector. Consider polytopes P ⊂ NR having edges
(1)
wρ = n−1
occur. Each such polytope defines a
ρ vρ with nρ ∈ N , where all ρ ∈ ∆
quotient presentation of the projective toric variety X associated to ∆:
ˆ the
Set N̂ = N ⊕ Z. Let σ̄ ⊂ N̂R be the cone generated by P × (0, 1), and ∆
fan of all strict faces σ̂ ( σ̄. Then the canonical projection Q : N̂ → N defines a
quotient presentation q : X̂ → X. In fact, these quotient presentations are precisely
those obtained from affine cones over X. A typical picture is Figure 2.
Figure 2. A quotient presentation of a projective toric surface
3. Enough effective Weil divisors
The goal of this section is to describe, up to isomorphism, the set of all quotient
presentations of a fixed toric prevariety X. Recall that we have a canonical map
div : M −→ WDivT (X),
m 7→ div(χm ),
where χm ∈ Γ(T, OX ) is the character function corresponding to m ∈ M . Suppose
q : X̂ → X is a quotient presentation. The inverse q∗ : WDivT̂ (X̂) → WDivT (X)
of the strict transform yields a factorization
M −→ M̂ −→ WDivT (X)
of the canonical map div : M → WDivT (X). We seek to reconstruct the quotient
presentation from such sequences.
Definition 3.1. A triangle is an abstract lattice M̂ , together with a sequence
M → M̂ → WDivT (X), such that the following holds: The composition is the
canonical map div : M → WDivT (X), the map M → M̂ is injective, and for each
invariant affine open subset U ⊂ X there is an m̂ ∈ M̂ whose image D ∈ WDivT (X)
is effective with support X \ U .
QUOTIENT PRESENTATIONS
5
Roughly speaking, the image of M̂ → WDivT (X) contains enough Weil divisors,
such that it generates the topology of X. Recall that a scheme Y is separated if
the diagonal morphism Y → Y × Y is a closed embedding. We say that Y is of
affine intersection if the diagonal is an affine morphism. In other words, there is
an affine open covering Ui ⊂ Y such that the Ui ∩ Uj are affine.
Theorem 3.2. Let X be a toric prevariety of affine intersection. For each quotient
presentation q : X̂ → X, the corresponding sequence M → M̂ → WDivT (X) is a
triangle. Up to isomorphism, this assignment yields a bijection between quotient
presentations and triangles.
Proof. Suppose q : X̂ → X is a quotient presentation. Given an invariant affine
open subset U ⊂ X, the preimage Û ⊂ X̂ is affine as well. There is an effective
invariant principal divisor D̂ ⊂ X̂ with support X̂ \ Û , because X̂ is quasiaffine.
So D = q∗ (D̂) is an effective Weil divisor with support X \ U . By construction,
D ∈ WDivT (X) lies in the image of M̂ .
φ
Conversely, suppose that M −→ M̂ −→ WDivT (X) is a triangle. Set for short
M̃ := WDivT (X). Let Ñ and N̂ denote the dual lattices of M̃ and M̂ respectively.
Dualizing the triangle, we obtain a sequence
ψ
Q
Ñ −→ N̂ −→ N.
For each prime divisor E ∈ M̃ , let E ∗ ∈ Ñ denote the dual base vector. For every
invariant open set U ⊂ X we have the submonoid Ñ+ (U ) generated by the E ∗ ,
where E ∈ WDivT (U ) is a prime divisor. Let σ̂U ⊂ N̂R be the cone generated by
ψ(Ñ+ (U )). For example, σ̂T = {0}.
We claim that σ̂U ⊂ σ̂X is a face provided that U ⊂ X is an affine invariant
open subset. Indeed by assumption, there is an m̂ ∈ M̂ such that D = φ(m̂) is an
effective Weil divisor with support X \ U . So for each prime divisor E ∈ M̃ , we
have
hψ(E ∗ ), m̂i = hE ∗ , φ(m̂)i = hE ∗ , Di ≥ 0,
with equality if and only if E ∗ ∈ Ñ+ (U ). So m̂ is a supporting hyperplane for σ̂X
cutting out σ̂U and the claim is verified. In particular, since σ̂T = {0} is a face of
σ̂X , this cone is strictly convex.
For later use, let us also calculate Q(ψ(E ∗ )). If vρ denotes the primitive lattice
vector in the ray ρ corresponding to the divisor E ∈ M̃ , we have
hQ(ψ(E ∗ )), mi = hE ∗ , divχm i = hvρ , mi .
That implies Q(ψ(E ∗ )) = vρ . So in particular, ψ(E ∗ ) is a primitive lattice vector,
and ψ induces a bijection between the rays of Ñ+ (X) and σ̂X .
ˆ be the fan in N̂ generated by the faces σ̂U , where U ⊂ X ranges over
Let ∆
all invariant affine open subsets. By construction, this defines a quasiaffine toric
variety X̂.
It remains to construct the quotient presentation q : X̂ → X. First, we do
this locally over an invariant affine open subset U ⊂ X. Let σU ⊂ NR be the
corresponding cone, and let Û ⊂ X̂ be the affine open subset defined by σ̂U .
Clearly, the map Q : N̂ → N has a finite cokernel, since M → M̂ was is assumed
to be injective. We have Q(σ̂U ) = σU . Moreover, it follows from we saw above
that the map Q ◦ ψ induces a bijection between the sets of primitive lattice vectors
6
A. A’CAMPO-NEUEN, J. HAUSEN, AND S. SCHROEER
generating the rays of Ñ+ (U ) and σU . Therefore the induced map vρ̂ → Q(vρ̂ ) gives
a bijection between the primitive lattice vectors generating the rays in σ̂U and σU .
By Proposition 2.3, the associated toric morphism Û → U is a quotient presentation. To obtain the desired quotient presentation q : X̂ → X, we glue the local
patches. Let U1 , U2 ⊂ X be two affine charts. The intersection U := U1 ∩ U2
is affine, and the rays of σU are in bijection with the invariant prime divisors in
U1 ∩ U2 . On the other hand, the rays of σ̂1 ∩ σ̂2 are the images of the duals to the
prime divisors in WDivT (U1 ∩ U2 ). This implies that Q(σ̂1 ∩ σ̂2 ) = σU .
¤
For the following examples, assume that X is a toric variety without nontrivial
torus factor. Equivalently, the map M → WDivT (X) is injective. Such toric
varieties are called nondegenerate.
id
Example 3.3. Obviously, the factorization M → WDivT (X) → WDivT (X) is a
triangle. The corresponding quotient presentation was introduced by Cox [4]. It is
the largest quotient presentation in the sense that it dominates all other nondegenerate quotient presentations of X.
Example 3.4. Suppose that for each invariant affine open subset U ⊂ X, the
complement X \ U is the support of an effective Cartier divisor. Then the factorization M → CDivT (X) → WDivT (X) is a triangle. The corresponding quotient
presentation q : X̂ → X was studied by Kajiwara [7]. He says that X has enough
Cartier divisors. Note that such toric varieties are divisorial schemes in the sense
of Borelli [1].
Example 3.5. Suppose X is a quasiprojective toric variety. Choose an ample
Cartier divisor D ∈ WDivT (X). Then M → M ⊕ ZD → WDivT (X) is a triangle.
The corresponding quotient presentation q : X̂ → X is nothing but the Gm -bundle
obtained from the vector bundle L → X associated to the ample sheaf OX (D).
Next, we come to existence of quotient presentations:
Proposition 3.6. A toric prevariety admits a quotient presentation if and only if
it is of affine intersection.
Proof. Suppose q : X̂ → X is a quotient presentation and consider two invariant
affine charts X1 , X2 of X. Since q is an affine toric morphism, the preimages
X̂i := q −1 (Xi ) are invariant affine charts of X̂.
The restriction of q defines a quotient presentation X̂1 ∩ X̂2 → X1 ∩ X2 . Since
X̂ is separated, the intersection X̂1 ∩ X̂2 is even affine. Property 2.3 (iii) implies
that the image X1 ∩ X2 = q(X̂1 ∩ X̂2 ) is again an affine toric variety.
Conversely, let X be of affine intersection. Choose a splitting M = M 0 ⊕ M 00 ,
where M 0 ⊂ M is the kernel of the canonical map M → WDivT (X). It suffices to
show that the canonical factorization
M −→ M 0 ⊕ WDivT (X) −→ WDivT (X)
is a triangle. Let U ⊂ X be an invariant affine open subset. We have to check that
the complement D = X \ U is a Weil divisor. For each invariant affine open subset
V ⊂ X, the intersection U ∩ V is affine, so V ∩ D is a Weil divisor. Hence D is a
Weil divisor.
¤
QUOTIENT PRESENTATIONS
7
4. Free and geometric quotient presentations
In this section we shall relate quotient presentations to geometric invariant theory. Fix a toric prevariety X, together with a quotient presentation q : X̂ → X
defined by a triangle M → M̂ → WDivT (X). Let G ⊂ T̂ be the kernel of the
induced homomorphism T̂ → T of tori. The question is: In what sense is X a
quotient of the G-action on X̂?
Note that G = Speck[W ], such that W = M̂ /M is the character group of the
group scheme G. Such group schemes are called diagonalizable. The G-action on
X̂ corresponds to a W -grading on
M
Rw
q∗ (OX̂ ) = R =
w∈W
for certain coherent OX -modules Rw . We call them the weight modules of the
quotient presentation. To describe the weight modules, consider the commutative
diagram
/ WDivT (X)
/ Cl(X)
/0
M
Q∗
²
M̂
q] '
²
/ WDivT̂ (X̂)
q]
²
/ Cl(X̂)
/ 0.
The snake lemma yields a map W → Cl(X). Hence each character w ∈ W gives an
isomorphism class of invariant reflexive fractional ideals:
Lemma 4.1. Each weight module Rw is an invariant reflexive fractional ideal.
The isomorphism class [Rw ] ∈ Cl(X) is the image of −w.
Proof. First, suppose that the quotient presentation q : X̂ → X is defined by an
inclusion of rings k[σ ∨ ∩ M ] ⊂ k[σ̂ ∨ ∩ M̂ ]. The weight module Rw ⊂ R is given by
the homogeneous component Rw ⊂ k[σ̂ ∨ ∩ M̂ ] of degree w ∈ W .
Let vρ ∈ N and vρ̂ ∈ N̂ be the primitive lattice vectors generating the rays in
σ (1) and σ̂ (1) , respectively. Choose m̂ ∈ M̂ representing w ∈ W . Note that the
T̂ -invariant Weil divisor q∗ (div(χm̂ )) on X̂ is given by the function
m̂ : σ (1) −→ Z,
ρ 7→ hm̂, vρ̂ i.
The reflexive fractional ideal R ⊂ k(X) over the ring k[σ ∨ ∩ M ] corresponding to
−[w] ∈ Cl(X) is generated by the monomials χm ∈ k[M ] with m ≥ −m̂ as functions
∗
on σ (1) . Obviously, the map χm 7→ χQ (m)+m̂ induces the desired bijection R →
Rw . This is compatible with localization, hence globalizes.
¤
Suppose a diagonalizable group scheme G acts on a scheme Y . An invariant
G
affine morphism f : T
Y → Z with
T OZ = f∗ (OY ) is called a good quotient. Note
that this implies f ( Wi ) = f (Wi ) for each family of invariant closed subsets
Wi ⊂ Y . Moreover, f : Y → Z is a categorial quotient.
Proposition 4.2. Each quotient presentation q : X̂ → X is a good quotient for the
G-action on X̂.
Proof. The problem is local, so we can assume that q : X̂ → X is given by an
inclusion of rings k[σ ∨ ∩ M ] ⊂ k[σ̂ ∨ ∩ M̂ ]. By Lemma 4.1, the ring of invariants
k[σ̂ ∨ ∩ M̂ ]G is nothing but k[σ ∨ ∩ M ].
¤
8
A. A’CAMPO-NEUEN, J. HAUSEN, AND S. SCHROEER
Sometimes we can do even better. Suppose a diagonalizable group scheme G acts
on a scheme Y . An invariant morphism f : Y → Z such that the corresponding
morphism G×Z Y → Y ×Z Y , (g, y) 7→ (gy, y) is an isomorphism is called a principal
homogeneous G-space. Equivalently, the projection Y → Z is a principal G-bundle
in the flat topology ([8] III Prop. 4.1).
Proposition 4.3. The quotient presentation q : X̂ → X is a principal homogeneous
G-space if and only if M̂ → WDivT (X) factors through the group of invariant
Cartier divisors.
T
Proof. Suppose that M̂ maps to CDiv
L (X). According to Lemma 4.1, the homogeneous components in q∗ (OX̂ ) = w∈W Rw are invertible. You easily check that
the multiplication maps Rw ⊗ Rw0 → Rw+w0 are bijective. So by [6] Proposition
4.1, the quotient presentation X̂ → X is a principal homogeneous G-space. Hence
the condition is sufficient. Reversing the arguments, you see that the condition is
necessary as well.
¤
Example 4.4. Regular toric prevarieties are factorial, hence their quotient presentations are principal homogeneous spaces. Consequently, an arbitrary quotient
presentation is a principal homogeneous space in codimension 1.
For the next result, let us recall another concept from geometric invariant theory.
Suppose a diagonalizable group scheme G acts on a scheme Y . A good quotient
Y → Z is called a geometric quotient if it separates the G-orbits.
Proposition 4.5. Suppose q : X̂ → X is a quotient presentation. Then X is a
geometric quotient for the G-action on X̂ if and only if M̂ → WDivT (X) factors
through the group of invariant Q-Cartier divisors.
Proof. First, we check sufficiency. Let M̂ 0 ⊂ M̂ be the preimage of the subgroup
CDivT (X) ⊂ WDivT (X). The group scheme H = Speck[M̂ /M̂ 0 ] is finite, so its
action on X̂ is automatically closed. Consequently, the quotient X̂ 0 = X̂/H is a
geometric quotient. You directly see that X̂ 0 is quasiaffine. Consider the induced
toric morphism q 0 : X̂ 0 → X. The strict transforms in
0
(q 0 )]
WDivT (X) −→ WDivT̂ (X̂ 0 ) −→ WDivT̂ (X̂)
are injective, and their composition is bijective. So the map on the right is bijective,
hence q 0 : X̂ 0 → X is another quotient presentation. By construction, its triangle
M → M̂ 0 → WDivT (X) factors through CDivT (X). According to Proposition
4.3, it is a geometric quotient. So q : X̂ → X is the composition of two geometric
quotients, hence a geometric quotient.
The condition is also necessary. Suppose X is a geometric quotient, that means
the fibers q −1 (x) are precisely the G-orbits. By definition, G acts freely on T̂ .
By semicontinuity of the fiber dimension, the stabilizers Gx̂ ⊂ G for x̂ ∈ X̂ must
be finite. Note that the stabilizers are constant along the T̂ -orbits. Hence the
stabilizers generate a finite subgroup H ⊂ G.
Set X̂ 0 = X̂/H. As above, we obtain a quotient presentation q 0 : X̂ 0 → X. By
construction, X is a free geometric quotient for the action of G0 = G/H. Now [9],
Proposition 0.9, ensures that q 0 : X̂ 0 → X is a principal homogeneous G0 -space. By
Proposition 4.3, the triangle M → M̂ 0 → WDivT (X) factors through CDivT (X).
QUOTIENT PRESENTATIONS
9
This implies that M̂ → WDivT (X) factors through the group of invariant Q-Cartier
divisors.
¤
Example 4.6. Simplicial toric varieties are Q-factorial, hence their quotient presentations are geometric quotients. It follows that arbitrary quotient presentations
are geometric quotients in codimension 2.
5. Homogeneous coordinates and multigraded modules
Throughout this section, fix a toric prevariety X of affine intersection and choose
a quotient presentation q : X̂ → X. The goal of this section is to relate quasicoherent OX -modules to multigraded modules over homogeneous coordinate rings. This
generalizes the classical approach for X = Pn , and results of Cox [4] and Kajiwara
[7] as well.
We propose the following definition of homogeneous coordinates. By assumption,
the toric variety X̂ is quasiaffine, so the affine hull X̄ = SpecΓ(X̂, OX̂ ) is an affine
toric variety. We call the ring S = Γ(X̂, OX̂ ) the homogeneous coordinate ring with
respect to the quotient presentation q : X̂ → X. Let M ⊂ M̂ → WDivT (X) be
its triangle and set W = M̂ /M . The action of the diagonalizable group scheme
G = Speck[W ] on L
X̂ induces a G-action on the affine hull X̄, which corresponds to
a W -grading S =
Sw .
Suppose F is a W -graded S-module. Then F corresponds to a quasicoherent
G-linearized OX̄ -module M. Let i : X̂ → X̄ be the open inclusion. The restriction
i∗ (M) is a G-linearized quasicoherent OX̂ -module. Because q : X̂ → X is affine,
this corresponds to a W -grading on
M
q∗ (i∗ (M)) =
q∗ (i∗ (M))w .
w∈W
∗
Definition 5.1. The sheaf F̃ = q∗ (i (M))0 is called the associated OX -module for
the W -graded S-module F .
For example, the OX -module associated to S is nothing but S̃ = OX . Clearly,
F 7→ F̃ is an exact functor from the category of W -graded S-modules to the
category of quasicoherent OX -modules. You easily check that the functor commutes
with direct limits and sends finitely generated modules to coherent sheaves.
We can pass from quasicoherent sheaves to graded modules
L as well. Suppose
F is a quasicoherent OX -module. Decompose q∗ (OX ) =
w∈W Rw into weight
L
modules. Then Γ∗ (F) = w∈W Γ(X̂, F ⊗OX Rw ) is a W -graded S-module.
Definition 5.2. We call Γ∗ (F) the W -graded S-module associated to the quasicoherent OX -module F.
For example, Γ∗ (OX ) = S. Obviously, F 7→ Γ∗ (F) is a functor from the category
of quasicoherent OX -modules to the category of W -graded S-modules.
Proposition 5.3. There is a canonical isomorphism F ' (Γ∗ (F))∼ for each quasicoherent OX -module F.
Proof. By definition, we have (Γ∗ (F))∼ = (q∗ i∗ i∗ q ∗ (F))0 . Since i : X̂ → X̄ is an
open embedding, i∗ i∗ (M) ' M holds for each quasicoherent OX̂ -module M. This
gives (q∗ i∗ i∗ q ∗ (F))0 ' (q∗ q ∗ (F))0 . Because R0 = OX , we have (q∗ q ∗ (F))0 ' F .
Consequently, F ' (Γ∗ (F))∼ .
¤
10
A. A’CAMPO-NEUEN, J. HAUSEN, AND S. SCHROEER
We see that the functor F 7→ F̃ from graded modules to quasicoherent sheaves is
surjective on isomorphism classes. It might happen, however, that F̃ = 0 although
F 6= 0. The next task is to understand the condition F̃ = 0. To do so, we first have
to generalize the classical notions of irrelevant ideals and Veronese subrings.
The reduced closed subset X̄ \ X̂ is an invariant closed subset inside the affine
toric variety X̄. We call the corresponding M̂ -homogeneous ideal S+ ⊂ S the
irrelevant ideal. Note that S+ = S holds if and only if X is affine.
Suppose U ⊂ X is an invariant open subset. Let WU ⊂ W be the subgroup of
all w ∈ W such that the corresponding invariant reflexive fractional ideal Rw is
invertible over U ⊂ X. Following a standard notation, we call the subring
M
M
S (WU ) =
Sw ⊂
Sw = S
w∈WU
w∈W
the Veronese subring with respect to WU L
⊂ W . Given a W -graded S-module F , we
have the Veronese submodule F (WU ) = w∈WU Fw as well. This is a WU -graded
S (WU ) -module.
Theorem 5.4. Suppose F is a finitely generated W -graded S-module. Then the
condition F̃ = 0 holds if and only if there is an invariant affine open covering
k
X = U1 ∪ . . . ∪ Un such that some power S+
⊂ S of the irrelevant ideal annihilates
(WU1 )
(WUn )
the Veronese submodules F
,...,F
.
Proof. Choose m̂i ∈ M̂ such that the homogeneous elements si = χm̂i ∈ S define
effective Weil divisors with support X \ Ui . Then I = (s1 , . . . , sn ) ⊂ S has the same
radical as the irrelevant ideal S+ ⊂ S.
Suppose that F̃ = 0. Then the restrictions Fi = F̃ | Ui are zero as well. Note
that the preimage Ûi = q −1 (Ui ) is affine, with global section ring Ssi . The Veronese
subring Si = (Ssi )(WUi ) defines a factorization
Ûi −→ Spec(Si ) −→ Ui .
According to Proposition 4.1, the map on the right is a principal bundle for the
action of the diagonalizable group scheme Gi = Speck[WUi ].
Setting Fi = (Fsi )(WUi ) , we conclude that the Si -module Fi is zero as well. Hence
ki
si · Fi = 0 for some integer ki > 0, because Fi is of finite type. Consequently
I k · Fi = 0 with k = max {ki }. This shows that the condition is necessary. The
converse is similiar.
¤
6. Morphisms into toric varieties
Throughout this section, fix a toric prevariety X of affine intersection. We seek to
describe the functor hX (Y ) = Hom(Y, X) represented by X in terms of sheaf data
on Y . Here Y ranges over the category of k-schemes. To do so, choose a quotient
presentation q : X̂ → X. Let S = Γ(X̂, OX̂ ) be the homogeneous coordinate ring
and set X̄ = Spec(S).
Given a k-scheme Y , we shall deal with pairs (A, ϕ) such that A is a W -graded
quasicoherent OY -algebra with A0 = OY , and ϕ : S ⊗ OY → A is a W -graded
homomorphism of OY -algebras. For simplicity, we refer to such pairs as S-algebras.
An S-algebra (A, ϕ) yields a diagram
QUOTIENT PRESENTATIONS
11
Spec(ϕ)
p
/Y
Spec(A)
X̄ × Y o
Â
Â
 r(A,ϕ)
Â
Â
²
²
²
q
/
X
X̄ o
X̂
The problem is to construct the dashed arrows. For this, we need a base-pointfreeness condition. Recall that the irrelevant ideal S+ ⊂ S is the ideal of the closed
subscheme X̄ \ X̂.
Definition 6.1. An S-algebra (A, ϕ) is called base-point-free if for each y ∈ Y
there is an M̂ -homogeneous s ∈ S+ such that the germ ϕ(s) := ϕ(s ⊗ 1) ∈ Ay is a
unit.
This is precisely what we need:
Proposition 6.2. Each base-point-free S-algebra (A, ϕ) defines, in a canonical
way, a morphism r(A,ϕ) : Y → X.
Proof. First, we claim that Spec(A) → X̄ factors through the open subset X̂ ⊂ X̄.
For y ∈ Y choose s ∈ S+ such that ϕ(s) is a unit in Ay . Then ϕ(s) is invertible on
a p-saturated neighbourhood of p−1 (y) ⊂ Spec(A). Clearly, this neighbourhood is
mapped into X̄s ⊂ X̂.
According to [9] Theorem 1.1, the projection Spec(A) → Y is a categorical quotient for the G-action defined by the W -grading on A (here we use the assumption
OY = A0 ). The composition Spec(A) → X̂ → X is G-invariant. So the universal
property of categorical quotients gives a commutative diagram
(6.2.1)
Spec(A)
/ X̂
²
Y
²
/ X,
which defines the desired morphism r(A,ϕ) : Y → X.
¤
Remark 6.3. The assignment (A, ϕ) 7→ r(A,ϕ) is functorial in the following sense:
Given a base-point-free S-algebra (A, ϕ) on Y and a morphism f : Y 0 → Y . Then
the preimage (A0 , ϕ0 ) = (f ∗ A, f ∗ ϕ) is a base-point-free S-algebra on Y 0 , and the
corresponding morphisms satisfy r(A0 ,ϕ0 ) = r(A,ϕ) ◦ f .
We call an M̂ -homogeneous element s ∈ S+ saturated, if X̂s = q −1 (q(X̂s )) holds.
In that case, Xs := q(X̂s ) is an affine invariant open subset with Γ(Xs , OX ) = S(s) .
Recall that X is covered by the sets Xs with s ∈ S+ saturated. We define Yϕ(s) ⊂ Y
to be the (open) subset of all y ∈ Y where the germ ϕ(s) ∈ Ay is a unit.
−1
(Xs ) for each
Lemma 6.4. With the preceding notation, we have Yϕ(s) = r(A,ϕ)
saturated s ∈ S+ .
Proof. Let y ∈ Yϕ(s) . Then ϕ(s) is invertible on a neighbourhood of the fibre of
Spec(A) → Y over y. Looking at the commutative diagram 6.2.1, we see that s is
invertible at some point of the fibre of q : X̂ → X over x := r(A,ϕ) (y). Since s is
saturated, this means x ∈ Xs . The reverse inclusion is clear by definition.
¤
12
A. A’CAMPO-NEUEN, J. HAUSEN, AND S. SCHROEER
Different base-point-free S-algebras may define the same morphism. To overcome
this, we need an equivalence relation. Suppose (A1 , ϕ1 ) and (A2 , ϕ2 ) are two basepoint-free S-algebras. Call them equivalent if for each saturated s ∈ S+ , say of
degree w ∈ W , the following holds:
(i) The open subsets Yϕi (s) ⊂ Y coincide for i = 1, 2.
(w)
(w)
(w)
(ii) Over Yϕ1 (s) = Yϕ2 (s) , the Ss -algebras A1 and A2 are isomorphic.
(w)
Here Ss
⊂ Ss is the Veronese subring with degrees in Zw ⊂ W .
Proposition 6.5. Two base-point-free S-algebras on Y define the same morphism
Y → X if and only if they are equivalent.
Proof. Suppose that (Ai , ϕi ) are two base-point-free S-algebras, which define two
morphisms ri : Y → X, with i = 1, 2. First, assume that r1 = r2 . Let s ∈ S+
be saturated. Using Lemma 6.4, we infer Yϕ1 (s) = Yϕ2 (s) . To check the second
condition for equivalence, note that
(w)
Ai |Yϕi (s) = OYϕi (s) [ϕi (s), ϕi (s)−1 ]
and Ss(w) = Γ(Xs , OX )[s, s−1 ]
are Laurent polynomial algebras. So the map ϕ1 (s) 7→ ϕ2 (s) induces the desired
isomorphism.
Conversely, assume that the base-point-free S-algebras are equivalent. Let s ∈
S+ be saturated, and let w ∈ W be its degree. Consider the partial quotients
(w)
Spec(Ai ) → Spec(Ai ) → Yϕi (s)
(w)
Then the isomorphism A2
the morphism
(w)
Spec(A1 )
(w)
→ A1
→
and
X̂s → Spec(Ss(w) ) → Xs
induces the identity on Yϕ1 (s) = Yϕ2 (s) . Thus
induces both, ri : Yϕi (s) → Xs .
¤
(w)
Spec(Ss )
We come to the main result of this section:
Theorem 6.6. The assignment (A, ϕ) 7→ r(A,ϕ) yields a functorial bijection between the set of equivalence classes of base-point-free S-algebras on Y and the set
of morphisms Y → X.
Proof. In Remark 6.3, we already saw that the assignment is functorial in Y . By
Proposition 6.5, it is well-defined on equivalence classes and gives an injection from
the set of equivalence classes to the set of morphisms. It remains to check that the
identity morphism id : X → X arises from a base-point-free S-algebra. Indeed: you
easily check that R = q∗ (OX̂ ), together with the adjunction map S ⊗ OX → R is
a base-point-free S-algebra defining the identity on X.
¤
As an application, we generalize the result of Kajiwara in [7]:
Proposition 6.7. Suppose the characteristic sequence M ⊂ M̂ → WDivT (X) of
the quotient presentation q : X̂ → X factors through the group of Cartier divisors.
Then two base-point-free S-algebras define the same morphism into X if and only
if they are isomorphic.
Proof. Let (A, ϕ) be a base-point-free S-algebra on the scheme Y defining a morphism r : Y → X. Set R = q∗ (OX̂ ). The map Spec(A) → X̂ defines a homomorphism R ⊗OX OY → A. Clearly, it suffices to show that this map is bijective.
The problem is local, so we may assume that X is affine, hence each weight module Rw ⊂ R is trivial and S+ = S holds. According to Lemma 6.4, for each
QUOTIENT PRESENTATIONS
13
M̂ -homogeneous unit s ∈ S, the image ϕ(s) ∈ Γ(Y, A) is a global unit. Since
each weight module Rw is generated by such a homogeneous unit, we infer that
R ⊗ OY → A is bijective.
¤
In general, the homogeneous components of a base-point-free S-algebra might
be noninvertible. However, this does not happen for quotient presentations that
are principal bundles:
Corollary 6.8. Assumptions as in Proposition 6.7. Then each base-point-free Salgebra (A, ϕ) has invertible homogeneous components Aw ⊂ A.
Proof. By assumption, R = q∗ (OX̂ ) has invertible homogeneous components. By
the preceding Proposition, each base-point-free S-algebra (A, ϕ) is isomorphic to
∗
(R).
¤
the preimage r(A,ϕ)
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[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
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Fachbereich Mathematik, Johannes Gutenberg-Universität, 55099 Mainz, Germany
E-mail address: acampo@enriques.mathematik.uni-mainz.de
Fachbereich Math. und Statistik, Universität Konstanz, 78457 Konstanz, Germany
E-mail address: Juergen.Hausen@uni-konstanz.de
Mathematische Fakultät, Ruhr-Universität, 44780 Bochum, Germany
E-mail address: s.schroeer@ruhr-uni-bochum.de